Post-Newtonian expansion of gravitational waves

1.2 Post-Newtonian expansion of gravitational waves

The post-Newtonian expansion of general relativity assumes that the internal gravity of a source is small so that the deviation from the Minkowski metric is small, and that velocities associated with the source are small compared to the speed of light, $c$ . When we consider the orbital motion of a compact binary system, these two conditions become essentially equivalent to each other. Although both conditions may be violated inside each of the compact objects, this is not regarded as a serious problem of the post-Newtonian expansion, as long as we are concerned with gravitational waves generated from the orbital motion, and, indeed, the two bodies are usually assumed to be point-like objects in the calculation.

In fact, Itoh, Futamase, and Asada [57 , 58 ] developed a new post-Newtonian method that can deal with a binary system in which the constituent bodies may have strong internal gravity, based on earlier work by Futamase and Schutz [44 , 45 , 42 ]. They derived the equations of motion to 2.5PN order and obtained a complete agreement with the Damour–Deruelle equations of motion [26 , 25 ], which assumes the validity of the point-particle approximation. In the Futamase–Schutz method, each star in a binary is first expressed as an extended object and then the limit is taken to set the radius to zero in a specific manner first proposed by Futamase [42 ]. At the same time, the surface integral approach (à la Einstein–Infeld–Hoffmann [32 ]) is taken to derive the equations of motion. More recently Itoh and Futamase [56 , 54 ] derived the 3PN equations of motion based on the Futamase–Schutz method, and they are again in agreement with those derived by Damour, Jaranowski and Schäfer [27 ] and by Blanchet et al. [10 ] in which the point-particle approximation is used. Update

There are two existing approaches of the post-Newtonian expansion to calculate gravitational waves: one developed by Blanchet, Damour, and Iyer (BDI) [12 , 7 ] and another by Will and Wiseman (WW) [111 ] based on previous work by Epstein, Wagoner, and Will [33 , 109 ]. In both approaches, the gravitational waveforms and luminosity are expanded in time derivatives of radiative multipoles, which are then related to some source multipoles (the relation between them contains the “tails”). The source multipoles are expressed as integrals over the matter source and the gravitational field. The source multipoles are combined with the equations of motion to obtain explicit expressions in terms of the source masses, positions, and velocities.

One issue of the post-Newtonian calculation arises from the fact that the post-Newtonian expansion can be applied only to the near-zone field of the source. In the conventional post-Newtonian formalism, the harmonic coordinates are used to write down the Einstein equations. If we define the deviation from the Minkowski metric as

the Einstein equations are schematically written in the form

together with the harmonic gauge condition, $μν ∂νh = 0$ , where $μν □ = η ∂μ∂ ν$ is the D’Alambertian operator in flat-space time, $μν η = diag (− 1,1,1,1)$ , and $μν Λ (h )$ represents the non-linear terms in the Einstein equations. The Einstein equations (2

) are integrated using the flat-space retarded integrals. In order to perform the post-Newtonian expansion, if we naively expand the retarded integrals in powers of $1∕c$ , there appear divergent integrals. This is a technical problem that arises due to the near-zone nature of the post-Newtonian approximation. In the BDI approach, in order to integrate the retarded integrals, and to evaluate the radiative multipole moments at infinity, two kinds of approximation methods are introduced. One is the multipolar post-Minkowski expansion, which can be applied to a region outside the source including infinity, and the other is the near-zone, post-Newtonian expansion. These two expansions are matched in the intermediate region where both expansions are valid, and the radiative multipole moments are evaluated at infinity. In the WW approach, the retarded integrals are evaluated directly, without expanding in terms of $1∕c$ , in the region outside the source in a novel way.

The lowest order of the gravitational waves is given by the Newtonian quadrupole formula. It is standard to refer to the post-Newtonian formulae (for the waveforms and luminosity) that contain terms up to $n 𝒪 ((v∕c) )$ beyond the Newtonian quadrupole formula as the ( $n ∕2$ )PN formulae. Evaluation of gravitational waves emitted to infinity from a compact binary system has been successfully carried out to the 3.5 post-Newtonian (PN) order beyond the lowest Newtonian quadrupole formula in the BDI approach [12 , 7 , 18 , 19 , 15 , 16 , 11 ]. Up to now, the WW approach gives the same answer for the gravitational waveforms and luminosity to 2PN order. Update

The computation of the 3.5PN flux requires the 3PN equations of motion. As mentioned in the above, the 3PN equations of motion have been derived by three different methods. The first is the direct post-Newtonian iteration in the harmonic coordinates [14 , 28 , 10 ]. The second employs the Arnowitt–Deser–Misner (ADM) coordinates within the Hamiltonian formalism of general relativity [59 , 60 , 27 ]. The third is based on the Futamase–Schutz method [56 , 54 ].

Since the first two methods use the point particle approximation while the third one is not, let us first focus on the first two. In both methods, since the stars are represented by the Dirac delta functions, the divergent self-fields must be regularized. In earlier papers, they used the Hadamard regularization method [59 , 60 , 14 , 28 ]. However, it turned out that there remains an unknown coefficient which cannot be determined within the regularization method. This problem was solved by Damour, Jaranowski and Schäfer [27 ] who successfully derived the 3PN equations of motion without undetermined numerical coefficients by using the dimensional regularization within an ADM Hamiltonian approach. Then the 3PN equations of motion in the harmonic coordinates were also derived without undetermined coefficients by using a combination of the Hadamard regularization and the dimensional regularization in [10 ]. The 3.5PN radiation reaction terms in the equations of motion are also derived in both approaches [76 , 62 ]. See reviews by Blanchet [8 , 9 ] for details and summaries on post-Newtonian approaches.

In the case of Futamase–Schutz method, as mentioned in the beginning of this subsection, the 3PN equations of motion is derived by Itoh and Futamase [56 , 54 ], and the 3.5PN terms are derived by Itoh [55 ]. See a review article by Futamase and Itoh [43 ] for details on this method.

There are other methods in which stars are treated as fluid balls [51 , 63 , 80 , 81 ]. Pati and Will [80 , 81 ] use an method which is an extension of the WW approach in which the retarded integral is evaluated directly. With these method, the 2PN equations of motion as well as 2.5PN and 3.5PN radiation reaction effects are derived.